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Probably not. Microsoft uses a one-way hash function to encrypt the key that's stored in the registry. I'm honestly surprised you were able to pull a 2013 product key, since it uses the same mechanism.

In short, unless you're Microsoft, you won't be able to retrieve it. You should be able to grab it from Microsoft if you linked it with an email address:

This person is a verified professional.

Even the last handful of digits from the product key is debatable. It could just be me, but I've never got the last few recovered characters to ever match up with a registered product... having said that, the online registration and key management process now seems such an ungodly disaster, I usually give up and put in an order for a new one.. I think it usually works out cheapr..